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20130126
20130203
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, a number times itself, as we know, is said to be "squared." three rows of three pebbles, 3 x 3, equaling 9. that's not a bad example. so most numbers, even if they couldn't be shaped into squares, could at least be represented by rectangles, where by "rectangle" we mean only those that have more than one pebble in each dimension. for example, take the number 12. it can be represented by rectangles that are either two rows of six pebbles or three rows of four pebbles, so that the height and width of these boxes actually give us a visualization of a basic multiplication problem. since a rectangle has length and width, the number of its pebbles represents both the rectangle itself and the answer to a multiplication problem such as 3 x 4. but notice that certain quantities of pebbles just don't fit in a box. if you're trying to make one of these so-called "fat" rectangles out of a number like 11 or 13, something is always left over. so a number like 11 or 13 can never be the answer to a multiplication problem other than 1 x itself. this ends up being true for many numbers: 2, 3, 5, 7, 11, 13,
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